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Demonstration

In math, x and y are two variables which people always use. If these x and y have some relations of ratio, is called proportions. The theory of proportion usually appears in our life. Such as when we save money in the bank and the interest is about proportion. And the tax is also about proportion. Generally speaking, proportion is used a lot in math area, especially in statistics that mathematician always use proportion to check, analyze, and compare different data so that they could make the right choice to better our society or reduce unnecessary consumption on whatever economics, energies, or faculties. What’s more, it can be stretched to other fields, such as computer science, astronomy, chemistry, and physics. For instance, the proportion is used in architecture, in order to build a stable building, the ratio and structure are important such as the golden number “ϕ = [(1 + √5)/2” In mathematics, proportion is the basic knowledge that could help us learn more in math. Such us the trigonometric function, we can calculate the angle if we know the other two sides in a right angled triangle. Or we can use one side and the angle between two sides to calculate the other side in triangle in a right angled triangle. So, the ratio is basically everywhere in our life. We may use ratio in our daily spending, calculating while we are doing math problems, designing clothes for people, or in some academic researches, such as the proportion of dosage on medical science, or consume on economics. Ratio is super essential basic knowledge that experts need to know for these advanced field and technologies. And we can see these kinds of knowledge everywhere during our life. The gold with every currency ratio is constant, and that’ s why the gold is worth and would not be devalued. That’ s why people are likely to fired gold in order to earn money by using the ratio of gold and currency.

Golden ratio

The ratio is not only used in math, physic, Art, Nature and human body, but also is popular used in architecture. Golden ratio is very famous which is used in a lot of buildings in order to make more beautiful buildings. For example, Egyptians used this data in order to build Great Pyramid in the past. 1.618 is also called Phi, and it also was called "Divine Proportion" in 1500. And the perfect ratio of people' s body is upper body divided by height equals 1.618. People nowadays believe golden section ratio was originated from Pythagorean school, which was consisted of group of natural scientists from ancient Greek in 500 B.C. In early 6th B.C, ancient Greek mathematician, Eudoxus, firstly did research on it and built a ratio theory. Leonardo da Vinci drew a picture of human body, which was a person in a circle with arms and feet touch the edge of circle. The shape of body looks more beautiful if people' s body ratio is applied in golden ratio.

Proportion in Statistics

Knowledge is based on our daily life and exceeded our life. Same as proportion in Statistics. It could be very big that applied in national wide population check, or it could be very small that applied in one single nail check. It formed normal distribution, binomial distribution and entirely percentage distribution to help us lot to effectively solve different types of math problems. Based on these proportions in statistics, people can estimate the trend by observing those data and make the right decision.

Constant Proportion

The ratio is always constant, which means two sides variables of one equation could transfer and remove to other sides in order to keep the ratio same. whenever these three variables are in which side, the proportion does not change. For example, y=kx, then we move k to the y side, k=y/x, so the y/x is always constant. Although the number times 2, 3 or more, the ratio is always constant. For example, y=3x, if we move x and y on the same side is y/x=3. In addition, if we move y to the right side, 1/3=x/y. No matter x and y in the same side or not, the ratio between them does not change and will keep constant. In addition, it was applied in Chemistry somehow. Constant proportion was once mentioned by French Chemist Joesph Proust that "A chemical compound always contains the same elements combined together in the same proportion by mass."

Directly Proportional Function

The function of directly proportion is y=kx + b, k=y/x, which k and b are constant, also y/x is constant, and the line of y/x on the graph should be a straight line, and then we could write it down as y ∝ x to show that y is proportional to x directly. If k is positive, the function line goes from 3rd quadrant to 1st quadrant. when x < 0, as x goes to negative infinity, then y also goes to negative infinity, which the line decreases gradually. As x approaches to zero, then y also approaches to 0. If k value is negative, the line of function goes from 2nd quadrant to 4th quadrant，.when x <0, y value goes to positive infinity which line increases gradually When x >0, y value goes to negative infinity. Whenever the x and y values positive or negative, k value is constant.

Inverse Proportional Function

y=k/x, which k is constant, also xy is constant. The line of xy is curved which represents the product of x and y is a constant value. The multiple of these two variables are constant. According to one value increases, the other one value decreases. For example, the total work is constant, one single person does it maybe he will spend much time. While if 10 people do it together, maybe the work time would reduce significantly. That’ s one specific example about inverse proportional function. When k is positive, these two lines show in the 1st quadrant and the 3rd quadrant. As x value increases, then y value decreases. X goes to the positive and negative infinity, y approaches to x axis (y not equal to 0). As x approaches to 0 (x not equal to 0), y goes to positive and negative infinity. On the other side, when k is negative, these two lines show in 2nd and 4th quadrant. As x value increases, y value also increases. As x goes to the positive and negative infinity, y approaches x axis gradually (y not equal to 0). As x approaches to 0 (x not equal to 0), y goes to positive and negative infinity. Because in the equation y=k/x denominator x cannot be 0, caused x cannot be, so 0 and y and k cannot be 0 otherwise this equation does not exist. Then x=0 is called x asymptote and y=0 is called y asymptote, which means that the function will never touch the x axis and y axis. Both of them will be close to these two axis, but never equals to 0.

Relevant Examples

Proportion in Statistics examples: 1. If people could know the proportion of getting a disabled baby depending on the time they plan to brood, then people can tell which date is less risky to born child so that they wouldn’t have a disabled baby. 2. If people could know the proportion of good quality of products produced by different factories, then they would make right choice so that they wouldn’t purchase inferior commodities, and it can somehow arouse producers and merchants produce good products to customers. Constant proportion examples: 1. When time is constant value, distance and speed are showing constant proportion. 2. When the area is constant value, the length and width are showing constant proportion. Directly proportional function examples: 1. The mass and volume of an object is directly proportional which has constant quantity of object’ s density. For example, the density of water is 1000kg/m^3. Force and area of an object is directly proportional which has constant quantity of object’ s pressure. P=W/A The base area and volume of a cylinder is directly proportional, because height of cylinder determines the ratio between V and Area of base. Height (cylinder)=V(cylinder)/ Area (cylinder base) Distance and time is directly proportional, because when velocity is constant, distance increases with the time increases. distance=velocity*time. If you work 1 hour and get 10 dollars, then when you work 2 hours, you will get 20 dollars. 2. If you have a steady speed of 2m/s walking on the street, then the distance and the time you spend on walking is proportional. It can be written as distance = speed * time. For instance, the time you spend on walking is 20 minutes, then the distance you travel should be 240 meters. 3. if you are chasing a car, which is 200 meters in front of you and has speed of 20m/s, and you are the best runner in the world that you have 40m/s speed. So, the time that you need to spend run over that car is 10 seconds. The formula for that is time = distance / (your speed – car’s speed), which is as same as x = y / k. 4. The currency ratio is also constant value, for example the currency between Australia and United State America is 1:0.694, which means that 1 Australian dollar is equal to 0.694 American dollar. So that is one kind of ratio between currency. And the ratio is not constant every day. If you want to exchange some foreign currency, checking the ratio and buy it when the ratio is low, therefore, you can spend less money in order to exchange more foreign currency. Inverse proportional function examples: If the total distance is constant. The higher velocity you have, the shorter time you will spend. Therefore, the velocity and time are inverse proportional if the total displacement does not change. 1. If the volume of cylinder keeps constant, the height of cylinder increases, the area of cylinder base should decrease in order to keep the constant of total volume. Therefore, the base area of cylinder and the height of cylinder is inverse proportional. 2. If you should eat 10 apples in 10 hours, but if you eat quicker and finish to eat 10 apples in one hour, then you save much time. Therefore, the speed of your eating and the time are inverse proportional function. 3. Or you have certain amount of money and decide to buy bunch of stationeries, then the numbers of stationeries and that of price are showing inverse proportion. To be more specific, if you have 30 dollars, then you cannot buy over 30 dollar stuff, because the total amount is finite, therefore, if you want to buy more stuff, you have to think about the price.

References

Michela Rossi, April 2014, Review of The Theory of Proportion in Architecture, Nexus Network Journal, Volume 16, Issue 1, pp 233–237 Eudoxus of Cnidus, 408BC, mathematician and astronomer, http://www-history.mcs.st-andrews.ac.uk/Biographies/Eudoxus.html Statistics How To, Stephanie, May 5th , 2016, https://www.statisticshowto.datasciencecentral.com/population-proportion/ www.baidu.image.com Joesph Proust, French Chemist, Proust’s Law of Constant Proportion firstly published in 1794. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Atomic_Theory/Dalton%27s_Atomic_Theory/Proust%27s_Law_of_Constant_Proportion Helmenstine, Anne Marie, Ph.D. "Direct Proportion Definition." ThoughtCo, Sep. 11, 2018, thoughtco.com/definition-of-direct-proportion-605034. Tomisti`/Wikimedia Commons CC BY SA UNSW Australia 2015. (https://www.futurelearn.com/courses/maths-linear-quadratic-relations/0/steps/1209) A. A. Goldstein C. Hobby， July 1971, 《Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg》，Volume 36, Issue 1, pp 136–139 Stapel, Elizabeth. "Finding the Inverse of a Function." Purplemath. Gary Meisner May 13, 2012, History of the Golden Ratio.